Short algebraic notation is now the commonest form of chess notation, probably because it is easiest to understand and looks more elegant.

One thing sabre23t left out was that the move number is an essential part of the notation. For example, white's move in move 4 in the game above would be 4. Qxf7++ (++ is checkmate, btw, there is no notation for double-check). To indicate Black's third move by itself, use three dots, as in 3. ... a5??

A single exclamation mark (!) denotes a very good move while a single question mark (?) indicates a blunder. A double exclamation mark (!!) is a rarely used symbol denoting truly brilliant chess moves (ala. Bobby Fischer) while a double question mark (??) indicates a gross blunder. A question mark followed by an exclamation mark (?!) is a move that is questionable but not an outright error while an exclamation mark followed by a question mark (!?) is a surprising move that appears risky.

One more thing - "e.p." is often used to indicate an en passant capture.

Descriptive notation is the old way of writing out chess games. In algebraic notation each square has one name but with descriptive they have two: one from white's point of view and the other from black's. This can make it kind of confusing. Black's perspective is in lower case only to make it easier to read.

Descriptive is written out piece name first, a dash and then the square the peice is moving to. For the Scholar's Mate descriptive would look like this.

1. P-K4, P-K4
2. B-QB4, P-KR3
3. Q-KR5, P-QR4
4. Q-KB7 mate

Moves are written out in the easiest way possible as long as only one move is legal. For instance, if you're moving a bishop you don’t have to explain which bishop you are moving since there is only one bishop that can be moved. If two rooks can go to Q6, one being on Q8 and the other on Q2, you would write it out R/2-Q6 or R/8-Q6.

Kills are denoted with an "x", for instance NxQ, for a knight killing the queen. If two knights can kill the queen it follows the same rules as above, N/2xQ. Checks are written "ch" and checkmate is "mate". Castling is the same as algebraic 0-0 or 0-0-0 for a queen side castle. When queening a pawn a bracket is used, P-Q8(Q).

This system of notation is still found in the classic chess books but fell out of favor because it was more confusing than algebraic.

Each type of piece is denoted by a single letter (P=pawn, N=knight,
B=bishop, R=rook, Q=queen, K=king). Although speakers of other
languages have different initials for the piece names (and despite
the worldwide governing organization (FIDE) being French),
I believe that "official" diagrams use the English letters.

White's pieces are denoted by upper-case letters, Black's by lower-case